Question 1

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 
-About the y-axis

Accepted Answer

A)  81/8  $\pi$ 
B)  27/2  $\pi$ 
C)  27/4  $\pi$ 
D)  81/4  $\pi$ 
A)  81/8  $\pi$ 
B)  27/2  $\pi$ 
C)  27/4  $\pi$ 
D)  81/4  $\pi$ B

Question 2

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the  
-y = 8x, y = 8, x = 0

Accepted Answer

A)  128 /3 $\pi$ 
B)  64/3 $\pi$ 
C)  4 $\pi$ 
D)  24$\pi$ 
A)  128 /3 $\pi$ 
B)  64/3 $\pi$ 
C)  4 $\pi$ 
D)  24$\pi$ A

Question 3

Find the volume of the solid generated by revolving the region about the y-axis.
-The region enclosed by the triangle with vertices ( 1, 0), ( 1, 2), ( 3, 2)

Accepted Answer

A)  26/3 $\pi$ 
B)  20/3 $\pi$ 
C)  10/3 $\pi$ 
D)  4/3 $\pi$ 
A)  26/3 $\pi$ 
B)  20/3 $\pi$ 
C)  10/3 $\pi$ 
D)  4/3 $\pi$ B

Question 4

Solve the problem.   
-A 25-kg mass is attached to a spring that hangs vertically and is stretched 1 m from the equilibrium position of the spring. Assume a linear spring with F(x) = kx. How much work is required to stretch the spring and lower the mass 1.5 m? Round to four decimal places when appropriate.

Accepted Answer

A)  459.375 J 
B)  1010.625 J 
C)  91.875 J 
D)  643.125 J 
A)  459.375 J 
B)  1010.625 J 
C)  91.875 J 
D)  643.125 J

Question 5

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the  
-y = x, y = 0, x = 3, x = 5

Accepted Answer

A)  98/3 $\pi$ 
B)  17 $\pi$ 
C)  8 $\pi$ 
D)  2/3  $\pi$ 
A)  98/3 $\pi$ 
B)  17 $\pi$ 
C)  8 $\pi$ 
D)  2/3  $\pi$

Question 6

Let R be the region bounded by the given curves. Find the volume of the solid generated when R is revolved about the given line.
-y =   + 1, y = 1, x = 0, and x = 2; about y = 1

Accepted Answer

A)  $\pi$  
B)  $\pi$  
C)  $\pi$  
D)  2$\pi$   
A)  $\pi$  
B)  $\pi$  
C)  $\pi$  
D)  2$\pi$

Question 7

Find the volume of the solid generated by revolving the shaded region about the given axis.  
-About the y-axis

Accepted Answer

A)  12 $\pi$ 
B)  3   - 6 $\pi$ 
C)  6   + 3 $\pi$ 
D)  6   - 12 $\pi$ 
A)  12 $\pi$ 
B)  3   - 6 $\pi$ 
C)  6   + 3 $\pi$ 
D)  6   - 12 $\pi$

Question 8

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method.
-The region bounded by y = 3x -   and y = x about the line x = 2

Accepted Answer

A)  4/3$\pi$ 
B)  2 $\pi$ 
C)  8/3 $\pi$ 
D)  4 $\pi$ 
A)  4/3$\pi$ 
B)  2 $\pi$ 
C)  8/3 $\pi$ 
D)  4 $\pi$

Question 9

Find the volume of the described solid. 
-The solid lies between planes perpendicular to the x-axis at   and   . The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the semicircle   to the semicircle   .

Accepted Answer

A)  72 
B)  18 
C)  144 
D)  36 
A)  72 
B)  18 
C)  144 
D)  36

Question 10

Find the area of the shaded region.          
-

Accepted Answer

A)  7/15 
B)  76/15 
C)  22/15 
D)  2 
A)  7/15 
B)  76/15 
C)  22/15 
D)  2

Question 11

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis.
-y =   , y = 0, x = 2, x = 4

Accepted Answer

A)  8$\pi$ 
B)  12 $\pi$ 
C)  24$\pi$ 
D)  16$\pi$ 
A)  8$\pi$ 
B)  12 $\pi$ 
C)  24$\pi$ 
D)  16$\pi$

Question 12

Solve the problem.   
-A 50-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and that the chain has a density of 10 kg/m. How much work is required to wind the chain onto the cylinder if a 50-kg block is attached to the end of the chain?

Accepted Answer

A)  27,500g J 
B)  25,050g J 
C)  15,000g J 
D)  12,550g J 
A)  27,500g J 
B)  25,050g J 
C)  15,000g J 
D)  12,550g J

Question 13

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis.
-y = 32 -   , y =   , x = 0

Accepted Answer

A)  64 $\pi$ 
B)  128$\pi$ 
C)  512 $\pi$ 
D)  256$\pi$ 
A)  64 $\pi$ 
B)  128$\pi$ 
C)  512 $\pi$ 
D)  256$\pi$

Question 14

Find the volume of the described solid. 
-The solid lies between planes perpendicular to the x-axis at   and   . The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the parabola   to the parabola   .

Accepted Answer

A)  245 
B)  2450 
C)  1225 
D)  2400 
A)  245 
B)  2450 
C)  1225 
D)  2400

Question 15

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 
-About the y-axis x = 2

Accepted Answer

A)  8/5  $\pi$ 
B)  16/3 $\pi$ 
C)  32/5  $\pi$ 
D)  16/5  $\pi$ 
A)  8/5  $\pi$ 
B)  16/3 $\pi$ 
C)  32/5  $\pi$ 
D)  16/5  $\pi$

Question 16

Solve the problem.   
-A cylinder with height 9 m and radius 3 m is filled with water and must be emptied through an outlet pipe 3 m above the top of the cylinder. Compute the work required to empty the water in the top half of the tank. Then compute the work required to empty the (equal amount of) water in the lower half of the tank. Round to two decimal places when appropriate.

Accepted Answer

A)  11,222,083.1 J; 5,611,041.55 J 
B)  5,611,041.55 J; 5,611,041.55 J 
C)  6,546,215.14 J; 12,157,256.7 J 
D)  18,703,471.8 J; 12,157,256.7 J 
A)  11,222,083.1 J; 5,611,041.55 J 
B)  5,611,041.55 J; 5,611,041.55 J 
C)  6,546,215.14 J; 12,157,256.7 J 
D)  18,703,471.8 J; 12,157,256.7 J

Question 17

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis.
-x = 4y -   , x = 0

Accepted Answer

A)  32/3 $\pi$ 
B)  128/3 $\pi$ 
C)  256/3 $\pi$ 
D)  64/3 $\pi$ 
A)  32/3 $\pi$ 
B)  128/3 $\pi$ 
C)  256/3 $\pi$ 
D)  64/3 $\pi$

Question 18

Find the volume of the solid generated by revolving the shaded region about the given axis.  
-About the y-axis

Accepted Answer

A)  16/3  $\pi$ 
B)  20/3  $\pi$ 
C)  32/5  $\pi$ 
D)  8/5  $\pi$ 
A)  16/3  $\pi$ 
B)  20/3  $\pi$ 
C)  32/5  $\pi$ 
D)  8/5  $\pi$

Question 19

Find the area enclosed by the given curves.
-Find the area of the region between the curve y = 2x/(1 +   ) and the interval   of the x-axis.

Accepted Answer

A)  2 ln 17 
B)  0 
C)  2   
D)  ln 17 
A)  2 ln 17 
B)  0 
C)  2   
D)  ln 17

Question 20

Solve the problem.   
-A 25-kg mass is attached to a spring that hangs vertically and is stretched 3 m from the equilibrium position of the spring. Assume a linear spring with F(x) = kx. How much work is required to compress the spring and lift the mass 1.5 m? Round to four decimal places when appropriate.

Accepted Answer

A)  275.625 J 
B)  398.125 J 
C)  30.625 J 
D)  643.125 J 
A)  275.625 J 
B)  398.125 J 
C)  30.625 J 
D)  643.125 J

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the y-axis

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the -y = 8x, y = 8, x = 0

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by the triangle with vertices ( 1, 0), ( 1, 2), ( 3, 2)

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the -y = x, y = 0, x = 3, x = 5

Let R be the region bounded by the given curves. Find the volume of the solid generated when R is revolved about the given line. -y = + 1, y = 1, x = 0, and x = 2; about y = 1

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the y-axis

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. -The region bounded by y = 3x - and y = x about the line x = 2

Find the volume of the described solid. -The solid lies between planes perpendicular to the x-axis at and . The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the semicircle to the semicircle .

Find the area of the shaded region. -

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. -y = , y = 0, x = 2, x = 4

Solve the problem. -A 50-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and that the chain has a density of 10 kg/m. How much work is required to wind the chain onto the cylinder if a 50-kg block is attached to the end of the chain?

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. -y = 32 - , y = , x = 0

Find the volume of the described solid. -The solid lies between planes perpendicular to the x-axis at and . The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the parabola to the parabola .

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the y-axis x = 2

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. -x = 4y - , x = 0

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the y-axis

Find the area enclosed by the given curves. -Find the area of the region between the curve y = 2x/(1 + ) and the interval of the x-axis.

Functions

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Derivatives

Applications of the Derivative

Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

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Exam 6: Applications of Integration

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the y-axis

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the -y = 8x, y = 8, x = 0

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by the triangle with vertices ( 1, 0), ( 1, 2), ( 3, 2)

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the -y = x, y = 0, x = 3, x = 5

Let R be the region bounded by the given curves. Find the volume of the solid generated when R is revolved about the given line. -y = + 1, y = 1, x = 0, and x = 2; about y = 1

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the y-axis

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. -The region bounded by y = 3x - and y = x about the line x = 2

Find the volume of the described solid. -The solid lies between planes perpendicular to the x-axis at and . The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the semicircle to the semicircle .

Find the area of the shaded region. -

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. -y = , y = 0, x = 2, x = 4

Solve the problem. -A 50-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and that the chain has a density of 10 kg/m. How much work is required to wind the chain onto the cylinder if a 50-kg block is attached to the end of the chain?

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. -y = 32 - , y = , x = 0

Find the volume of the described solid. -The solid lies between planes perpendicular to the x-axis at and . The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the parabola to the parabola .

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the y-axis x = 2

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. -x = 4y - , x = 0

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the y-axis

Find the area enclosed by the given curves. -Find the area of the region between the curve y = 2x/(1 + ) and the interval of the x-axis.

Functions

Limits

Derivatives

Applications of the Derivative

Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Calculus

Filters